Ten students recorded the number of hours, h, they spent revising for a test and their mark, m. The scatter diagram shows the results.
Describe the correlation between h and m.
Interpret this correlation in the context of the question.
State which variable is the explanatory variable.
3 marks
Show answer
a) positive correlationb) students who revised for longer tended to get higher marksc) h, the number of hours spent revising
2
A café owner records the daily maximum temperature, x °C, and the number of ice creams sold, y, on 20 days in summer. The temperatures ranged from 14 °C to 28 °C.
The equation of the regression line of y on x is
y = 12.5 + 3.2x
Give an interpretation of the gradient of the regression line.
Estimate the number of ice creams sold on a day when the maximum temperature is 20 °C.
Comment on the reliability of using the line to estimate the number sold on a day when the maximum temperature is 35 °C.
4 marks
Show answer
a) for each 1 °C rise in maximum temperature, about 3.2 more ice creams are soldb) 76 or 77 (76.5)c) unreliable: 35 °C is outside the range of the data (14 °C to 28 °C), so this is extrapolation
3
For each pair of variables, state which is the explanatory variable and which is the response variable, and state whether you would expect positive or negative correlation.
The amount of fertiliser given to a tomato plant and the mass of tomatoes it produces.
The outside temperature and the amount of gas a house uses for heating each day.
4 marks
Show answer
a) explanatory: amount of fertiliser; response: mass of tomatoes; positive correlationb) explanatory: outside temperature; response: gas used; negative correlation
4
A council collects data for 15 towns in a region. For each town it records the number of cafés and the number of road traffic accidents last year. The data show strong positive correlation.
A councillor says, “Cafés cause road accidents, so we should stop new cafés opening.”
Explain why the councillor’s conclusion is not justified.
Suggest a variable that could explain the correlation.
3 marks
Show answer
a) correlation does not imply causation: the two variables are linked but one does not have to cause the otherb) the population of the town (larger towns have more cafés and more traffic)
5
The table shows the engine size, x litres, and the fuel economy, y miles per gallon, of eight cars.
Engine size, x (litres)
1.0
1.2
1.4
1.5
1.8
2.0
2.5
3.0
Fuel economy, y (mpg)
56
54
51
50
46
43
37
31
The equation of the regression line of y on x is y = 68.9 − 12.7x
State, giving a reason, which variable is the explanatory variable.
Give an interpretation of the gradient of the line.
Estimate the fuel economy of a car with a 1.6 litre engine.
Explain why the value 68.9 has no practical meaning here.
Explain why the line should not be used to estimate the engine size of a car that does 40 mpg.
5 marks
Show answer
a) engine size, x: the engine size affects the fuel economyb) for each extra litre of engine size, fuel economy falls by about 12.7 mpgc) 48.6 mpg (3 s.f.)d) it is the estimate for an engine size of 0 litres, which is impossible (and far outside the data)e) the line is y on x, so it is only designed to estimate y from a given x
6
The table shows the age, x years, and the value, y thousand pounds, of eight cars of the same model sold by a garage.
Age, x (years)
1
2
3
4
5
6
7
8
Value, y (£1000s)
16.5
14.0
12.2
10.1
15.8
7.0
5.6
4.1
On the grid, draw a scatter diagram of these data.
One point does not fit the pattern. Write down its coordinates and suggest a reason for it.
Describe the correlation shown by the other seven points.
4 marks
Show answer
a) eight crosses plotted at the points in the tableb) (5, 15.8): e.g. that car had very low mileage or was in excellent condition, or the value was recorded wronglyc) strong negative correlation
7
A researcher measures the foot length, f cm, and the reading score, r, of 14 children: seven from Year 3 and seven from Year 9. The scatter diagram shows the results.
Describe the correlation shown by all 14 points.
Kofi says, “Children with longer feet are better readers.” With reference to the scatter diagram, comment on Kofi’s claim.
Describe the correlation within each year group.
4 marks
Show answer
a) positive correlationb) the points form two separate groups, Year 3 and Year 9; the older children have both longer feet and higher scores, so age explains the pattern, not foot lengthc) little or no correlation in each year group
8
Lewis uses data from a UK weather station for 31 days in May. He records the daily total sunshine, s hours, and the daily maximum temperature, t °C. The values of s range from 0 to 14.2
The regression line of t on s is t = 13.4 + 0.62s
State the units of the gradient and give an interpretation of the gradient.
Give an interpretation of the value 13.4
Estimate the maximum temperature on a day with 8 hours of sunshine.
The sunshine for one day was recorded as n/a. Explain how Lewis should deal with this day before finding the regression line.
6 marks
Show answer
a) °C per hour: each extra hour of sunshine is linked with a maximum temperature about 0.62 °C higherb) the estimated maximum temperature on a day with no sunshine is 13.4 °Cc) 18.4 °C (3 s.f.)d) n/a means the value is not available, so that day should be removed from the data
9
A taxi firm models the cost, £y, of a journey of x miles by the regression line
y = 2.80 + 1.65x
Give an interpretation of the values 2.80 and 1.65
The firm decides to record distances, k, in kilometres, where 1 km = 0.621 miles.
Find the equation of the regression line of y on k, giving the gradient to 3 significant figures.
Use your equation to estimate the cost of a 12 km journey.
5 marks
Show answer
a) £2.80 is the fixed charge (cost of a journey of 0 miles); £1.65 is the extra cost for each mileb) y = 2.80 + 1.02kc) £15.04 (accept £15.10 if the unrounded gradient is used)
10
The scatter diagram shows the speed, x mph, and the stopping distance, y metres, of a car in six tests. The regression line of y on x is
y = −27.4 + 1.69x
Describe the correlation between x and y.
Give an interpretation of the gradient of the line.
With reference to the scatter diagram, state, giving a reason, whether a linear regression model is suitable for these data.
Use the line to estimate the stopping distance at 10 mph, and comment on your answer.
6 marks
Show answer
a) strong positive correlationb) each extra 1 mph of speed adds about 1.69 m to the stopping distancec) not suitable: the points lie on a curve that gets steeper, not a straight lined) -10.5 m; a distance cannot be negative, and 10 mph is outside the data, so the line is not valid there
11
An ice cream seller works at two sites. She models her daily takings, £y, against the daily maximum temperature, x °C.
Site
Regression line of y on x
Temperatures in the data
Beach
y = −120 + 22x
15 °C to 30 °C
Park
y = 40 + 9x
12 °C to 28 °C
Give an interpretation of the value 22 in the beach model.
Find the temperature at which the two models predict the same takings.
Comment on the reliability of your answer to part (b).
The forecast maximum temperature for tomorrow is 26 °C. Find how much more the models predict she would take at the beach than at the park.
7 marks
Show answer
a) at the beach each 1 °C rise in maximum temperature is linked with about £22 more takingsb) 12.3 °C (3 s.f.)c) unreliable: 12.3 °C is below the beach data (15 °C to 30 °C), so it is extrapolation for the beach modeld) £178
12
A school nurse records the height, h cm, and the mass, m kg, of 25 fourteen-year-olds. The heights range from 148 cm to 182 cm. She codes the data using
x = h − 1505 and y = m − 40
The regression line of y on x is y = 4.2 + 0.85x
Find the equation of the regression line of m on h in the form m = a + bh
Give an interpretation of the value of b.
Estimate the mass of a fourteen-year-old who is 170 cm tall.
The nurse uses the line to estimate the mass of an adult who is 195 cm tall. Give two reasons why this estimate may be unreliable.
8 marks
Show answer
a) m = 18.7 + 0.17hb) each extra 1 cm of height is linked with about 0.17 kg more massc) 47.6 kgd) 195 cm is outside the range of heights in the data (extrapolation); the data are for fourteen-year-olds, not adults
Every question here is original and written by Robinson Tuition. It follows the Edexcel AS specification and suits AQA and OCR AS maths too; this is not an exam-board paper.